# What is the name of this inequality involving sums of squares and rectangles?

• Noesis
In summary, there does not seem to be a specific popular name for this inequality, but similar inequalities such as the rearrangement inequality and the Cauchy-Riemann-Schwartz inequality have been mentioned in relation to it. The inequality involves comparing the sum of squares to the sum of rectangles, which can also be understood geometrically. It is also related to vector spaces and the dot product.
Noesis
It's a trivial inequality...but I am curious if there is a popular name for it already. I have not been able to find it through conventional online search.

a12 + a22 + ... + an2 > a1*a2 + a2*a1 + a2*a3 + ... + an*an-1

Such as:

a2 + b2 + c2 > ab + ac + bc + ba + bc + ac = 2(ab + bc + ac)

Which can nicely be envisioned geometrically as sum of squares will always be bigger than corresponding sum of rectangles.

Noesis said:
It's a trivial inequality...but I am curious if there is a popular name for it already. I have not been able to find it through conventional online search.

a12 + a22 + ... + an2 > a1*a2 + a2*a1 + a2*a3 + ... + an*an-1

Such as:

a2 + b2 + c2 > ab + ac + bc + ba + bc + ac = 2(ab + bc + ac)

Which can nicely be envisioned geometrically as sum of squares will always be bigger than corresponding sum of rectangles.

It doesn't look like it's true. Let a=b>c. Then a2+b2=2ab, while c2<2(ac+bc).

Noesis said:
Which can nicely be envisioned geometrically as sum of squares will always be bigger than corresponding sum of rectangles.
There are a lot more of the rectangles than there are the squares... in fact, another easy source of counterexamples is just to make all the numbers the same! e.g. if they're all 1, then the L.H.S. is n, but the R.H.S. is n(n-1)

A very similar inequality is called the rearrangement inequality which works when you rearrange two sets of number, not one.

Noesis said:
It's a trivial inequality...but I am curious if there is a popular name for it already. I have not been able to find it through conventional online search.

a12 + a22 + ... + an2 > a1*a2 + a2*a1 + a2*a3 + ... + an*an-1

Such as:

a2 + b2 + c2 > ab + ac + bc + ba + bc + ac = 2(ab + bc + ac)

Which can nicely be envisioned geometrically as sum of squares will always be bigger than corresponding sum of rectangles.

Short tutorial on the subject.

First to get it right you need to omit half the terms.
a2 + b2 + c2 >= ab + bc + ac
There is a general theorem for vector spaces with inner (dot) products, namely
A.B=|A||B|cosx, where x is the angle between A and B.

For the case you are interested in:
A=(a1,a2,a3,...,an) while B=(a2,a3,...,an,a1)

mathman said:
Short tutorial on the subject.

First to get it right you need to omit half the terms.
a2 + b2 + c2 >= ab + bc + ac
There is a general theorem for vector spaces with inner (dot) products, namely
A.B=|A||B|cosx, where x is the angle between A and B.

For the case you are interested in:
A=(a1,a2,a3,...,an) while B=(a2,a3,...,an,a1)

I had a homework assignment in my first year of undergrad asking for us to prove the Cauchy-Riemann-Schwartz inequality (a.k.a. the triangle inequality), and Googling one or both of these terms seem to generate at least a few results that have a similar form.

## 1. What is an inequality?

An inequality is a mathematical statement that compares two quantities or expressions using symbols such as <, >, ≤, or ≥. It represents a relationship between two values that are not equal.

## 2. Why is it important to name an inequality?

Naming an inequality helps us to better understand and communicate its meaning. It also allows us to easily refer to and solve the inequality in future calculations or equations.

## 3. How do you determine the direction of an inequality?

The direction of an inequality is determined by the location of the variables or expressions in relation to the inequality symbol. For example, if the variable is on the left side of the symbol, the inequality will read "less than" or "greater than." If the variable is on the right side, the inequality will read "greater than" or "less than."

## 4. What are some examples of inequalities?

Some examples of inequalities include: x > 5, 2x + 3 ≤ 10, 3y < x + 2, and 4z ≥ 20. These all represent different relationships between two values, with one being larger or smaller than the other.

## 5. How can you solve an inequality?

To solve an inequality, we use similar methods to solving equations, such as adding, subtracting, multiplying, or dividing both sides by the same number. However, when multiplying or dividing by a negative number, the direction of the inequality symbol must be flipped. The solution to an inequality is a range of values that make the inequality true.

### Similar threads

• Precalculus Mathematics Homework Help
Replies
39
Views
5K
• General Math
Replies
1
Views
3K
• Calculus and Beyond Homework Help
Replies
2
Views
5K
• MATLAB, Maple, Mathematica, LaTeX
Replies
1
Views
2K
• Precalculus Mathematics Homework Help
Replies
4
Views
15K
• Precalculus Mathematics Homework Help
Replies
4
Views
3K
• STEM Academic Advising
Replies
10
Views
4K
• MATLAB, Maple, Mathematica, LaTeX
Replies
5
Views
2K
• MATLAB, Maple, Mathematica, LaTeX
Replies
7
Views
2K
• MATLAB, Maple, Mathematica, LaTeX
Replies
9
Views
2K